Carmichael's totient function conjecture: Difference between revisions

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In mathematics, '''Carmichael's Totient Function Conjecture''' concerns the [[multiplicity]] of values of [[Euler's totient function]] φ(''n''), the function which counts the number of integers less than and [[coprime]] to ''n''.

This conjecture asserts that there exist natural numbers ''m'', ''n'', ''m≠n'', for which <math>\varphi(m)=\varphi(n)</math>, where <math>\varphi()</math> denotes the [[Euler's totient function]].

This function &phi;(''n'') is equal to 2 when ''n'' is one of the three values 3, 4, and 6. It is equal to 4 when ''n'' is one of the four values 5, 8, 10, and 12. It is equal to 6 when ''n'' is one of the four values 7, 9, 14, and 18. In each case, there is more than one value of ''n'' having the same value of &phi;(''n'').

The conjecture asserts that this phenomenon of repeated values holds for every ''n''. That is, for every ''n'' there is at least one other integer ''m''&nbsp;≠&nbsp;''n'' such that &phi;(''m'')&nbsp;=&nbsp;&phi;(''n'').

==References==